Fractional Burgers Equation Research Articles

Finite Element Method on locally refined composite meshes for Dirichlet fractional Laplacian

It is known that the solution of the Dirichlet fractional Laplacian in a bounded domain exhibits singular behavior near the boundary. Consequently, numerical discretizations on quasi-uniform meshes lead to low accuracy and nonphysical solutions. We adopt a finite element discretization on locally refined composite meshes, which consist in a combination of graded meshes near the singularity and uniform meshes where the solution is smooth. We also provide a reference strategy on parameter selection of locally refined composite meshes. Numerical tests confirm that finite element method on locally refined composite meshes has higher accuracy than uniform meshes, but the computational cost is less than that of graded meshes. Our method is applied to discrete the fractional-in-space Allen–Cahn equation and the fractional Burgers equation with Dirichlet fractional Laplacian, some new observations are discovered from our numerical results.

Novel Approach by Shifted Fibonacci Polynomials for Solving the Fractional Burgers Equation

This paper analyzes a novel use of the shifted Fibonacci polynomials (SFPs) to treat the time-fractional Burgers equation (TFBE). We first develop the fundamental formulas of these polynomials, which include their power series representation and the inversion formula. We establish other new formulas for the SFPs, including integer and fractional derivatives, in order to design the collocation approach for treating the TFBE. These derivative formulas serve as tools that aid in constructing the operational metrics for the integer and fractional derivatives of the SFPs. We use these matrices to transform the problem and its underlying conditions into a system of nonlinear equations that can be treated numerically. An error analysis is analyzed in detail. We also present three illustrative numerical examples and comparisons to test our proposed algorithm. These results showed that the proposed algorithm is advantageous since highly accurate approximate solutions can be obtained by choosing a few terms of retained modes of SFPs.

Pseudospectral analysis for multidimensional fractional Burgers equation based on Caputo fractional derivative

This study presents the Chebyshev pseudospectral approach in time and space to approximate a solution to the time-fractional multidimensional Burgers equation. The suggested approach utilizes Chebyshev–Gauss–Lobatto (CGL) points in both spatial and temporal directions. To figure out the fractional derivative matrix at CGL points, we use the Caputo fractional derivative formula. Further, the Chebyshev fractional derivative matrix is utilized to reduce the given problem in an algebraic system of equations. The numerical approach known as the Newton–Raphson is implemented to get the desired results for the system. Error analysis for the set of values of ν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \ u $$\\end{document} is done for various model examples of fractional Burgers equations, where ν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ u $$\\end{document} represents the fractional order. The computed numerical results are in perfect agreement with the exact solutions.

Atomic Solution for Fractional Non-linear Burgers Equation of Conformable Type

Atomic Solution for Fractional Non-linear Burgers Equation of Conformable Type

An Efficient Cubic B-Spline Technique for Solving the Time Fractional Coupled Viscous Burgers Equation

The second order Burger’s equation model is used to study the turbulent fluids, suspensions, shock waves, and the propagation of shallow water waves. In the present research, we investigate a numerical solution to the time fractional coupled-Burgers equation (TFCBE) using Crank–Nicolson and the cubic B-spline (CBS) approaches. The time derivative is addressed using Caputo’s formula, while the CBS technique with the help of a θ-weighted scheme is utilized to discretize the first- and second-order spatial derivatives. The quasi-linearization technique is used to linearize the non-linear terms. The suggested scheme demonstrates unconditionally stable. Some numerical tests are utilized to evaluate the accuracy and feasibility of the current technique.

Modified hybrid B-spline estimation based on spatial regulator tensor network for burger equation with nonlinear fractional calculus

In this paper, a modified hybrid B-spline approximation based on spatial regulator tensor network (MHBA-SRTN) is proposed for solving the application and the feasibility problems of fractional calculus-based Burger equation. The main innovation points include: (1) a dual singular kernel based fractional derivative operator is employed on the Burger’s equation with external force term to analyze the solutions and properties under perturbation condition for further realism simulation; (2) a modified hybrid B-spline basis function and its corresponding topological tensor network structure are further proposed to solve the provided Burger’s equation with more favorable approximation accuracy and solution uniqueness; (3) the tensor network transformed the hybrid B-spline construction process into the tensor solving with the introduced spatially regulated tensor network decomposition for a more robust solution procedure; (4) an acceleration approach for decomposition process is introduced to significantly reduce its computational and storage complexity. Extensive experiments are also carried out to verify the performance of MHBA-SRTN.

Analytical solution to fractional differential equation arising in thermodynamics

The analysis of nonlinear events related to physical phenomena is a popular issue in the modern day. The essential purpose of this work is to discover a novel approximate solution to the fractional nonlinear Benjamin Bona Mahony Peregrine Burgers equation (BBMPB) utilizing the natural decomposition method (NDM) of fractional order. The suggested approach provides analytical solutions that are extremely near to the exact solution obviating the complexities associated with many other approaches. The expected issue’s uniqueness theorem and convergence analysis are explored using Banach’s fixed-point theory. The reliability and accuracy of the recommended method were tested using numerical simulations. The graphs and tables reflect the results. The comparison of the suggested scheme’s solution with the exact solutions demonstrates that the scheme is efficient, methodical, and extremely exact in tackling nonlinear complicated phenomena.

Solution of nonlinear fractional partial differential equations by Shehu transform and Adomian decomposition method (STADM)

Several well-known nonlinear partial differential equations (NPDEs) of integer order, including the Heat Equation and the Korteweg–de Vries (KDV) Equation, as well as NPDEs of fractional order, including the fractional advection equation (FAE), fractional gas dynamic equation (FGDE), diffusion equation, and fractional Burger equation (FBE), are approximated analytically in this study using the Shehu Transform Adomian Decomposition Method, which is a combination of two methods, namely, Shehu Transform (ST) and Adomian Decomposition Method (ADM). The Caputo sense is utilized to the fractional derivatives. This semianalytical method is used to acquire the series solutions to several example problems. The main advantage of the approach is that, in contrast to other similar semianalytical methods, it is straightforward to apply. This method has been widely used to answer any class of equations in the sciences and engineering because it can solve a huge class of linear and nonlinear equations effectively, more quickly, and precisely. The obtained solution matches exactly with the solution obtained by other methods.

Two linear finite difference schemes based on exponential basis for two-dimensional time fractional Burgers equation

In present paper, we propose tailored finite point method (TFPM) based on exponential basis for solving two-dimensional time fractional Burgers equation. We use the L1 and L1-2 discretization formulas for the Caputo fractional derivative in time direction and use the TFPM to approximate the spatial derivatives. This method with the help of exponential functions fit the properties of the local solution in time and space simultaneously which act as basis functions in the frame of TFPM. We focus on constructing two linear implicit finite difference schemes, i.e. a five-point node centered scheme and a four-point cell-centered scheme based on exponential functions for deriving the numerical solution of given problem. The stability and convergence of fully discrete schemes based on five-point node centered are discussed. At last, we use some numerical examples to show the accuracy and efficiency of the presented schemes.

Probabilistic global well-posedness to the nonlocal Degasperis–Procesi equation

We study the almost sure existence of a global solution of the nonlocal Degasperis–Procesi equation. After a suitable randomization, we obtain the local existence and uniqueness of the mild solution for a large set of the supercritical initial data. Then, we establish an energy estimate of the solution and get the global existence in probability. The methods can be used to get the well-posedness of fractional Burgers equation with the supercritical initial data.

A Potent Collocation Approach Based on Shifted Gegenbauer Polynomials for Nonlinear Time Fractional Burgers’ Equations

This paper presents a numerical strategy for solving the nonlinear time fractional Burgers's equation (TFBE) to obtain approximate solutions of TFBE. During this procedure, the collocation approach is used. The proposed numerical approximations are supposed to be a double sum of the products of two sets of basis functions. The two sets of polynomials are presented here: a modified set of shifted Gegenbauer polynomials and a shifted Gegenbauer polynomial set. Some specific integers and fractional derivatives are explicitly given as a combination of basis functions to apply the proposed collocation procedure. This method transforms the governing boundary-value problem into a set of nonlinear algebraic equations. Newton's approach can be used to solve the resulting nonlinear system. An analysis of the precision of the proposed method is provided. Various examples are presented and compared to some existing methods in the literature to prove the reliability of the suggested approach.

Coupled Fixed Point and Hybrid Generalized Integral Transform Approach to Analyze Fractal Fractional Nonlinear Coupled Burgers Equation

In this manuscript, the nonlinear Burgers equations are studied via a fractal fractional (FF) Caputo operator. The results of coupled fixed point theorems in cone metric space are used to discuss the uniqueness of solution to the proposed coupled equations. The solution of the proposed equation is computed via Natural transform associated with the Adomian decomposition method (NADM). The acquired results are graphically presented for some values of fractional order and fractal dimensions. The accuracy and consistency of the applied method is verified through error analysis.

Analytical solutions and mathematical simulation of traveling wave solutions to fractional order nonlinear equations

We investigate the tanh–coth method’s usefulness in understanding space–time fractional nonlinear evolution problems. To use this method and then examine their plots, we take into consideration three significant equations: the space–time fractional Burgers equation, the space–time fractional regularized long wave equation, and the space–time fractional Boussinesq equation. The modified Riemann–Liouville derivatives are used to define the fractional derivative. The approach provides a linearization- or small perturbation-free analytical solution in the form of a convergent series with easily calculable components. For the aforementioned nonlinear fractional equations, we have discovered their precise solutions. We utilize a generalized fractional complex transform to translate these to ordinary differential equations which subsequently resulted into number of exact solutions. The method provides us abundant analytical traveling wave solutions containing fewer number parameters with the aid of Mathematica and MATLAB.

Compatible extension of the [formula omitted]-expansion approach for equations with conformable derivative

In this study, the compatible extensions of the (G′/G)-expansion approach and the generalized (G′/G)-expansion scheme are proposed to generate scores of radical closed-form solutions of nonlinear fractional evolution equations. The originality and improvements of the extensions are confirmed by their application to the fractional space-time paired Burgers equations. The application of the proposed extensions highlights their effectiveness by providing dissimilar solutions for assorted physical forms in nonlinear science. In order to explain some of the wave solutions geometrically, we represent them as two- and three-dimensional graphs. The results demonstrate that the techniques presented in this study are effective and straightforward ways to address a variety of equations in mathematical physics with conformable derivative.

Efficient Spectral Collocation Method for Tempered Fractional Differential Equations

Transient anomalous diffusion may be modeled by a tempered fractional diffusion equation. In this paper, we present a spectral collocation method with tempered fractional Jacobi functions (TFJFs) as basis functions and obtain an efficient algorithm to solve tempered-type fractional differential equations. We set up the approximation error as O(Nμ−ν) for projection and interpolation by the TFJFs, which shows “spectral accuracy” for a certain class of functions. We derive a recurrence relation to evaluate the collocation differentiation matrix for implementing the spectral collocation algorithm. We demonstrate the effectiveness of the new method for the nonlinear initial and boundary problems, i.e., the fractional Helmholtz equation, and the fractional Burgers equation.

Fractional Burgers equation with singular initial condition

We consider the fractional Burgers equation ut=Δα/2u+b⋅∇(u|u|(α−1)/β) on Rd, d≥2, with α∈(1,2) and β>1 and prove the existence of a solution for a large class of initial conditions, which contains functions that do not belong to any Lp(Rd), 1≤p≤∞. Next, we apply the general results to the initial condition u0(x)=M|x|−β, 1<β<d, and show the existence of a selfsimilar solution and derive its properties such as smoothness, two-sided estimates, asymptotics and gradient estimates.

Solution of the Modified Time Fractional Coupled Burgers Equations Using Laplace Adomian Decompostion Method

Abstract In this work, a coupled system of time-fractional modified Burgers’ equations is considered. Three different fractional operators: Caputo, Caputo-Fabrizio and Atangana-Baleanu operators are implemented for the equations. Also, two different scenarios are examined for each fractional operator: when the initial conditions are u(x, y, 0) = sin(xy), v(x, y, 0) = sin(xy), and when they are u(x, y, 0) = e{−kxy}, v(x, y, 0) = e{−kxy}, where k, α are some positive constants. With the aid of computable Adomian polynomials, the solutions are obtained using Laplace Adomian decomposition method (LADM). The method does not need linearization, weak nonlinearity assumptions or perturbation theory. Simulations are also presented to support theoretical results, and the behaviour of the solutions under the three different fractional operators compared.

Spatio-temporal fractional shock waves solution for fractional Korteweg-de Vries burgers equations

ABSTRACT We examine the nonlinear characteristics of localized spatial (temporal) fractional shock excitations in a viscous fluid. By applying the natural transform decomposition method and with the aid of standard Caputo's operators, the stationary shock solutions for the spatial (temporal) fractional Korteweg-de Vries Burgers equations are derived. The results reveal that the temporal (Spatial) fractional index significantly modifies the localized solutions for the shocks. Moreover, The Banach contraction analysis shows that the wave solutions converge at , where stands for the temporal (Spatial) variables. Numerical analysis confirms that the index α effect nonlinear waves coupling and therefore causes the steeping of the fractional shocks. Variations in γ on the other hand reduce the wave dispersion and in turn localized the shock potentials. This study is important to understand the nonlinear evolution of shock waves in collisional plasma where the spatial/temporal fractional shocks suffer modifications in the pulse amplitude as well as in the spatial extensions.

A new approximate method to the time fractional damped Burger equation

&lt;abstract&gt;&lt;p&gt;In this article, we study a Caputo fractional model, namely, the time fractional damped Burger equation. As the main mathematical tool of this article, we apply a new approximate method which is called the approximate-analytical method (AAM) to deal with the time fractional damped Burger equation. Then, a new approximate solution of this considered equation was obtained. It may be used to characterize nonlinear phenomena of the shallow water wave phenomena. Thereby, it provides a new window for us to find the time fractional damped Burger equation new evolutionary mechanism.&lt;/p&gt;&lt;/abstract&gt;

Variational Iteration Method for Solving Time Fractional Burgers Equation Using Maple

The Time Fractional Burger equation was solved in this study using the Mabel software and the Variational Iteration approach. where a number of instances of the Time Fractional Burger Equation were handled using this technique. Tables and images were used to present the collected numerical results. The difference between the exact and numerical solutions demonstrates the effectiveness of the Mabel program’s solution, as well as the accuracy and closeness of the results this method produced. It also demonstrates the Mabel program’s ability to quickly and effectively produce the numerical solution.